Gujarat Board GSEB Textbook Solutions Class 11 Maths Chapter 1 Sets Ex 1.6 Textbook Questions and Answers.
Gujarat Board Textbook Solutions Class 11 Maths Chapter 1 Sets Ex 1.6
Question 1.
If X and Y are two sets such that n(X) = 17, n(Y) = 23 and n(X ∪ Y) = 38, find n(X ∩ Y).
Solution:
Since, n(X ∪ Y) = n(X) + n(Y) – n(X ∩ Y), so
38 = 17 + 23 – n(X ∩ Y)
⇒ n(X ∩ Y) = 17 + 23 – 38
= 40 – 38
= 2.
Question 2.
If X and Y are two sets such that X ∪ Y has 18 elements, X has 8 elements and Y has 15 elements, how many elements does X ∩ Y have?
Solution:
Here n(X) = 8, n(Y) = 15 and n(X ∪ Y) = 18.
Now, n(X ∪ Y) = n(X) + n(Y) – n(X ∩ Y).
⇒ 18 = 8 + 15 – n(X ∩ Y)
⇒ n(X ∩ Y) = 8 + 15 – 18
= 5.
Question 3.
In a group of 400 people, 250 can speak Hindi and 200 can speak English. How many people can speak both Hindi and English?
Solution:
Let H and E denote the sets of people who can speak Hindi and English respectively.
Then, n(H) = 250, n(E) = 200 and n(H ∪ E) = 400.
Now, n(H ∩ E) = n(H) + n(E) – n(H ∪ E)
= 250 + 200 – 400
= 50.
Thus, 50 people can speak both Hindi and English.
Question 4.
If S and T are two sets such that S has 21 elements, T has 32 elements and S ∩ T has 11 elements, how many elements does S ∪ T have?
Solution:
Here n(S) = 21, ra(T) = 32 and n(S ∩ T) = 11.
Now, n(S ∪ T) = n(S) + n(T) – n(S ∩ T)
= 21 + 32 – 11
= 53 – 11
= 42.
Question 5.
If X and Y are two sets such that X has 40 elements, X ∪ Y has 60 elements and X ∩ Y has 10 elements, how many elements does Y have?
Solution:
Here n(X) = 40, n(X ∪ Y) = 60, n(X ∩ Y) = 10, n(Y) = ?
Now, n(X ∪ Y) = n(X) + n(Y) – n(X ∩ Y).
⇒ 60 = 40 + n(Y) – 10
⇒ n(Y) = 60 – 40 + 10
= 20 + 10
= 30.
∴ Y has 30 elements.
Question 6.
In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many like both coffee and tea?
Solution:
Let C be the set of people who like coffee and T be Lie set of people who like tea.
Then, n(C ∪ T) = 70, n(C) = 37, n(T) = 52.
We know that
n(C ∪ T) = n(C) + n(T) – n(C ∩ T).
∴ 70 = 37 + 52 – n(C ∩ T)
⇒ n(C ∩ T) = 37 + 52 – 70
= 89 – 70
= 19.
∴ 19 people like both coffee and tea.
Question 7.
In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?
Solution:
Let C be the set of people who like cricket and T be the set of people who like tennis.
Then, n(C ∪ T) = 65, n(C) 40, n(C ∩ T) = 10.
We know that
n(C ∪ T) = n(C) + n(T) – (C ∩ T)
So, 65 = 40 + n(T) – 10
⇒ n(T) = 65 – 40 + 10 = 35.
∴ Number of people who like only tennis
= n(T) – n( C ∩ T)
= 35 – 10 = 25.
∴ Number of people who like tennis only and not cricket = 25 and number of people who like tennis is 35.
Question 8.
In a committee, 50 people speak French, 20 speak Spanish and 10 speak both Spanish and French. How many speak at least one of these two languages?
Solution:
Let F be the set of people who speak French and S be the set of people who speak Spanish.
Then, n(F) = 50, n(S) = 20 and n(F ∩ S) = 10.
We know that:
n(F ∪ S) = n(F) + n(S) – n(F ∩ S)
= 50 + 20 – 10
= 70 – 10
= 60.
∴ 60 people speak at least one of these two languages.